such that $\src\circ\iota=\trgt\circ\iota=\mathrm{id}_{G_0}$. In particular, the map $1_{-}$ is a monomorphism. The same vocabulary as for categories is used : elements of $G_0$ are \emph{objects} or \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells}, arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A \emph{morphism of reflexive graphs}$ f : G \to G'$ consists of maps $f_0 : G_0\to G'_0$ and $f_1 : G_1\to G'_1$ that commute with sources, targets and units in an obvious sense. This defines the category $\Rgrph$ of reflexive graphs. There is a ``underlying reflexive graph'' functor

\[

U : \Cat\to\Rgrph,

\]

which has a left adjoint

\[

L : \Rgrph\to\Cat.

\]

For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of $G$ and an arrow $f$ of of $L(G)$ is a chain

of composable arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer $n$ is referred to as the \emph{length} of $f$. Composition is given by concatenation of chains.

\end{paragr}

\begin{lemma}

A category $C$ is free in the sense of \todo{ref} if and only if there exists a reflexive graph $G$ such that

\[

C \simeq L(G).

\]

\end{lemma}

\begin{proof}

If $C$ is free, consider the reflexive graph $G$ such that $G_0= C_0$ and $G_1$ is the subset of $C_1$ whose elements are either generating $1$-cells of $C$ or units. It is straightforward to check that $C\simeq L(G)$.

Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the description of the arrows of $L(G)$ given in the previous paragraph shows that $C$ is free and that its set of generating $1$-cells is (isomorphic to) the non unital $1$-cells of $G$.

\end{proof}

\begin{remark}

Note that for a morphism of reflexive graphs $ f : G \to G'$, the functor $L(f)$ is not necessarily rigid in the sense of \todo{ref} because generating $1$-cells may be sent to units.

\end{remark}

\begin{paragr}

There is another important description of the category $\Rgrph$. Consider $\Delta_{\leq1}$ the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$. Then, the category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq1}}$, the category of pre-sheaves on $\Delta_{\leq1}$.